Advanced Calculus Single Variable

7.4 Continuous And Nowhere Differentiable

How bad can it get in terms of a continuous function not having a derivative at some points?
It turns out it can be the case the function is nowhere differentiable but everywhere
continuous. An example of such a pathological function different than the one I am about to
present was discovered by Weierstrass in 1872. Before showing this, here is a simple
observation.

Lemma 7.4.1Suppose f′

(x)

exists and let c be a number. Then letting g

(x)

≡ f

(cx)

,

g′(x) = cf′(cx).

Here the derivative refers to either the derivative, the left derivative, or the right derivative.Also, if f

(x)

= a + bx, then

′
f (x) = b

where again, f′refers to either the left derivative, right derivative or derivative. Furthermore,in the case where f

(x )

= a + bx,

f (x+ h)− f (x) = bh.

Proof: It is known from the definition that

′
f (x+ h)− f (x )− f (x) h = o(h)

Therefore,

g(x+ h)− g(x) = f (c(x+ h))− f (cx) = f ′(cx)ch+ o (ch)

and so

g (x + h)− g(x)− cf′(cx )h = o(ch) = o (h )

and so this proves the first part of the lemma. Now consider the last claim.

f (x+ h)− f (x) = a + b(x+ h)− (a+ bx) = bh
= bh + 0 = bh +o (h ).

Thus f′

(x)

= b. ■

Now consider the following description of a function. The following is the graph of the
function on

[0,1]

.

PICT

The height of the function is 1/2 and the slope of the rising line is 1 while the slope of the
falling line is −1. Now extend this function to the whole real line to make it periodic of period
1. This means f

(x+ n)

= f

(x)

for all x ∈ ℝ and n ∈ ℤ, the integers. In other words to find
the graph of f on

[1,2]

you simply slide the graph of f on

[0,1]

a distance of 1 to get the
same tent shaped thing on

[1,2]

. Continue this way. The following picture illustrates
what a piece of the graph of this function looks like. Some might call it an infinite
sawtooth.

PICT

Now define

∑∞ ( 3)k ( k )
g(x) ≡ 4 f 4 x .
k=0

Letting Mk =

(3∕4)

−k, an application of the Weierstrass M test shows g is everywhere
continuous. This is because each function in the sum is continuous and the series converges
uniformly on ℝ.